# Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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### Stable module category of non-Frobenius algebras

It is often said that the stable module category $A-\underline{\operatorname{mod}}$ for an associative algebra $A$ is triangulated if $A$ is Frobenius (i.e. over $A$ we have projective = injective). ...

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### Does a fully faithful and essentially surjective exact functor between triangulated categories have a quasi-inverse the 2-cat of triangulated cats?

$\def\D{\mathcal{D}}
\def\I{\mathcal{I}}
\def\A{\mathcal{A}}$Triangulated categories are the objects of a 2-category $\mathsf{Triang}$: the 1-morphisms are the exact functors $(F,\xi)$ of triangulated ...

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### The Balmer spectrum and the thick tensor ideals of the derived category of a Hopf algebra

Given a Hopf algebra $H$ over a field $\mathbb{k}$, the category of finite-dimensional left-$H$-modules naturally becomes a rigid monoidal category with exact monoidal product. Thus clearly the ...

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### What is the most general notion of exactness for functors between triangulated categories?

For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not ...

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### A particular morphism being zero in the singularity category

Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...

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### Thick subcategory containment in bounded derived category vs. singularity category

Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...

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### Grothendieck group and an almost localization

Let $T$ be a small triangulated category and let $S\subset T$ be a full triangulated subcategory. We denote this embedding by $I: S\rightarrow T$.
Let $F: T\rightarrow S$ be a triangulated functor ...

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### Example of triangulated category with vanishing $K_0$

Let $R$ be a ring, let $\operatorname{Perf}(R)$ the category of perfect modules over $R$. Suppose we have $E$ an perfect $R$-module (concentrated in degree $0$) such that its class $[E]\in K_0(R)$ is ...

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### Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?

$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...

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### Image, upto direct summands, of derived push-forward of resolution of singularities

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...

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### Exact sequences in Positselski's coderived category induce distinguished triangles

I am learning about Positselski's co- and contraderived categories. We know that short exact sequences do not generally induce distinguished triangles in the homotopy category but they do in the usual ...

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### Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...

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### When would a left admissible triangulated subcategory be admissible

I'm walking through the proof of [1, Thm 16 at pp. 515] and am stuck at the first sentence after equation (12), where the author states that the decomposition (12) is semiorthogonal when $a\geq 0$. ...

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### From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences

Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...

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### Why is this map a split monomorphism?

I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence.
It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R)
Let $Y$ be an object of the derived ...

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### When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories.
Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...

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### What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category).
Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...

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### Examples of tensor-triangulated categories not satisfying the local-to-global principle

From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...

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### Multiplication map by a ring element on an object vs. all its suspensions in singularity category

Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...

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### Vanishing of self-hom in Spanier–Whitehead stabilization category

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...

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### Compatibility of different exchange structures $\operatorname{Ex}^*_{\#},\operatorname{Ex}^*_*,\operatorname{Ex}_{\# *}$

Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...

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### structure in triangulated category similar to t-structure

It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with ...

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### Which "tensor" endofunctors on triangulated categories are essentially exact?

Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send ...

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### Slope-stability, tilt-stability, and Bridgeland stability

Following the definition of slope-stability ($\mu$-stability) and tilt-stability ($\nu$-stability) on page 8 of https://arxiv.org/abs/1410.1585, does an object's tilt-stability imply its slope-...

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### How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?

I am starting to learn the $K$-theory of triangulated categories and is stuck with the following.
Let $\mathcal{T}$ be a triangulated category having a semi-orthogonal decomposition $\langle \mathcal{...

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### Direct images commute with homotopy colimits

In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...

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### "Essential injectivity" of Balmer spectra

Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt-)category. Balmer defined a functor from the category of tt-categories to the category of locally ringed spaces, called the Balmer spectra or tt-...

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### Triangulated structure on complexes of mixed Hodge structures

I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures"
One aspect I don't really understand is the construction of the ''mixed cone'' for ...

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### Moral reason for negative sign in rotation axiom for triangulated categories

I would like to know if there is a "moral" reason why in the definition of triangulated categories the "rotation axiom" TR2 requires that we have to add a negative sign to an arrow ...

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### Can higher G-theory of Noetherian schemes be computed by derived categories?

Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K_0({\mathcal A})\cong K_0(D^{b}(\mathcal A))$.
When we set $\mathcal A$ to ...

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### When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?

Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\...

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### Why the stable module category?

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...

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### Generators of triangulated category and Grothendieck groups

Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and ...

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### Literature request: $K^b(\text{proj} A)$ Krull-Schmidt for $\text{gl dim}A = \infty$ and general results about its Grothendieck group

I'm interested in the Grothedieck group of the triangulated category $K^b(\text{proj}A)$ when $A$ is a finite dimensional algebra over a field of infinite global dimension.
For this purpose, It would ...

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### Decompose an unbounded (cochain) complex in the homotopy category

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...

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### Mapping cone is a functor

It is a well-known general fact that in a triangulated category, the cone $Z$ of a morphism $X \longrightarrow Y$ (that means there exists a distinguished triangle $X \longrightarrow Y \longrightarrow ...

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### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...

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### Autoequivalence group from semiorthogonal decomposition

Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{...

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### Can the Picard-graded homotopy of a nonzero object be nilpotent?

Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...

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### Did anybody study split homotopy cartesian squares in triangulated categories?

Let us call a commutative square
$$ \require{AMScd}
\begin{CD}
A @>{g'}>> B \\
@V{f'}VV @VV{f}V \\
C @>>{g}> D
\end{CD}
$$
in a triangulated category split homotopy ...

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### Rotation axiom for the triangulation on a derivator

I'm having some trouble following an argument in Moritz Groth's paper on Derivators, pointed derivators and stable derivators. More precisely, I'm currently stuck on the rotation axiom of the ...

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### Monoidal triangulated categories

I have a monoidal (not symmetric) triangulated category $(A,\otimes, 1)$ with unit 1.
Define $C$ the localizing subcategory of $A$ generated by the unit 1.
is $(C, \otimes, 1) $ a symmetric monoidal ...

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### Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?

A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...

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### Grothendieck group of triangulated categories

Let $A$ be a full triangulated subcategory of $B$, $u:A\rightarrow B$ the corresponding embedding. Let $f:B\rightarrow A$ be a triangulated functor
satisfying:
$f\circ u = id$
Let $b \in B $, if $f(b)...

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### Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...

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### Classification of 2-periodic triangulated categories

Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...

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### When is an object-preserving autoequivalence isomorphic to the identity?

Consider a triangulated category $\mathcal{T}$ and an exact autoequivalence $\Phi:\mathcal{T}\rightarrow \mathcal{T}$ such that $\Phi(F)=F$ for any object $F$ in $\mathcal{T}$, when could we say that $...

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### What is the Balmer spectrum of the p-complete stable homotopy category?

When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...

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### How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent?

Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see The classification of triangulated subcategories) that the ...

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### Is the intersection of two compactly generated localizing subcategories still compactly generated?

Suppose we have a compactly generated triangulated category $\mathcal{T}$ such that the subcategory of compact objects $\mathcal{T}^c$ is essentially small. Let us take $\mathcal{A}, \mathcal{B}$ two ...